Properties

Label 2-1840-5.4-c1-0-45
Degree $2$
Conductor $1840$
Sign $0.984 + 0.172i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.25i·3-s + (0.386 − 2.20i)5-s − 1.44i·7-s − 7.62·9-s + 2.03·11-s − 0.557i·13-s + (7.17 + 1.25i)15-s − 3.91i·17-s − 6.73·19-s + 4.70·21-s + i·23-s + (−4.70 − 1.70i)25-s − 15.0i·27-s + 9.69·29-s + 3.07·31-s + ⋯
L(s)  = 1  + 1.88i·3-s + (0.172 − 0.984i)5-s − 0.545i·7-s − 2.54·9-s + 0.612·11-s − 0.154i·13-s + (1.85 + 0.325i)15-s − 0.950i·17-s − 1.54·19-s + 1.02·21-s + 0.208i·23-s + (−0.940 − 0.340i)25-s − 2.89i·27-s + 1.80·29-s + 0.552·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.984 + 0.172i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.984 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434945494\)
\(L(\frac12)\) \(\approx\) \(1.434945494\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.386 + 2.20i)T \)
23 \( 1 - iT \)
good3 \( 1 - 3.25iT - 3T^{2} \)
7 \( 1 + 1.44iT - 7T^{2} \)
11 \( 1 - 2.03T + 11T^{2} \)
13 \( 1 + 0.557iT - 13T^{2} \)
17 \( 1 + 3.91iT - 17T^{2} \)
19 \( 1 + 6.73T + 19T^{2} \)
29 \( 1 - 9.69T + 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 + 3.65iT - 37T^{2} \)
41 \( 1 - 7.03T + 41T^{2} \)
43 \( 1 + 5.06iT - 43T^{2} \)
47 \( 1 + 0.659iT - 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 1.73T + 61T^{2} \)
67 \( 1 + 12.0iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 9.26iT - 73T^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + 5.25T + 89T^{2} \)
97 \( 1 + 0.0813iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.172002978828924845509847491006, −8.790518100387555695827680769740, −8.034995833279724749218268511076, −6.66118764963117000560056841490, −5.72903369967559171427337947455, −4.87916280385220381448373060877, −4.34897666600668510329530432561, −3.70122764085551656922093456148, −2.46402197389089594655461471035, −0.57262405666992691812698506406, 1.22312984020402347322105115605, 2.29038338860313140629543203680, 2.83176528100669412459336396096, 4.22275390293608620202969110671, 5.77906458288743794623223085708, 6.38046045323592082135533685270, 6.67407615713014886595005666992, 7.61735613253926166618031651093, 8.385958010387751859616165763879, 8.884437011927465732523874682482

Graph of the $Z$-function along the critical line